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A New Geometric Morita Invariant for Higher Rank Graph $C^*$-algebras

Mackenzie Amann, Liam Gallagher, Rachael Norton, Efren Ruiz

TL;DR

The paper extends the geometric classification program to higher rank graphs by introducing LiMaR-split, a neighborhood-based outsplit move that preserves Morita equivalence under a mild sink-free assumption. It establishes a stronger result: $C^*(\Lambda)$ is stably $*$-isomorphic to a corner of $C^*(\Gamma)$ with diagonal preservation and compatibility with the $k$-torus gauge actions, implying conjugacy of the associated $\mathbb{N}^k$-dynamics. The algebraic avenue via Kumjian-Pask algebras shows a $\,\mathbb{Z}^k$-graded corner isomorphism, which lifts to a diagonal-preserving $*$-isomorphism of the $C^*$-algebras after stabilization. Together, these results provide a robust invariant move for higher rank graph classification and connect the geometric and $K$-theoretic perspectives through stable equivalence and diagonal data.

Abstract

Higher rank graphs, also known as $k$-graphs, are a $k$-dimensional generalization of directed graphs and a rich source of examples of $C^*$-algebras. In the present paper, we contribute to the geometric classification program for $k$-graph $C^*$-algebras by introducing a new move on $k$-graphs, called LiMaR-split, which is a generalization of outsplit for directed graphs. We show, under one additional assumption, that LiMaR-split preserves the $k$-graph $C^*$-algebras up to Morita equivalence.

A New Geometric Morita Invariant for Higher Rank Graph $C^*$-algebras

TL;DR

The paper extends the geometric classification program to higher rank graphs by introducing LiMaR-split, a neighborhood-based outsplit move that preserves Morita equivalence under a mild sink-free assumption. It establishes a stronger result: is stably -isomorphic to a corner of with diagonal preservation and compatibility with the -torus gauge actions, implying conjugacy of the associated -dynamics. The algebraic avenue via Kumjian-Pask algebras shows a -graded corner isomorphism, which lifts to a diagonal-preserving -isomorphism of the -algebras after stabilization. Together, these results provide a robust invariant move for higher rank graph classification and connect the geometric and -theoretic perspectives through stable equivalence and diagonal data.

Abstract

Higher rank graphs, also known as -graphs, are a -dimensional generalization of directed graphs and a rich source of examples of -algebras. In the present paper, we contribute to the geometric classification program for -graph -algebras by introducing a new move on -graphs, called LiMaR-split, which is a generalization of outsplit for directed graphs. We show, under one additional assumption, that LiMaR-split preserves the -graph -algebras up to Morita equivalence.

Paper Structure

This paper contains 7 sections, 22 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Higher rank graphs
  4. Higher rank graph $C^*$-algebras
  5. Kumjian-Pask algebras
  6. LiMaR-split for higher rank graphs
  7. Proving Morita equivalence of $C^*(\Lambda)$ and $C^*(\Gamma)$

Key Result

Theorem 2.1

Hazlewoodefgggp If $G$ is a $k$-colored directed graph equipped with a degree map $d: G^* \to \mathbb{N}^k$ and range and source maps $r,s: G^1 \to G^0$, then $(\Lambda = G^* /\sim,d)$ is a $k$-graph for any $(r,s,d)$-preserving equivalence relation $\sim$ on $G^*$ which also satisfies the following Note that if $\lambda\sim\mu$ in a $k$-graph, we say that $\lambda$commutes with $\mu$.

Theorems & Definitions (61)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 2.1
  • Definition 5
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Definition 6
  • ...and 51 more